We propose a new model for natural image statistics. Instead of minimizing dependency between components of natural images, we maximize a simple form of dependency in the form of tree-dependency. By learning filters and tree structures which are best suited for natural images we observe that the resulting filters are edge filters, similar to the famous ICA on natural images results. Calculating the likelihood of the model requires estimating the squared output of pairs of filters connected in the tree. We observe that after learning, these pairs of filters are predominantly of similar orientations but different phases, so their joint energy resembles models of complex cells.

We study learning formulations with non-convex regularizaton that are natural for sparse linear models. There are two approaches to this problem: (1) Heuristic methods such as gradient descent that only find a local minimum. A drawback of this approach is the lack of theoretical guarantee showing that the local minimum gives a good solution. (2) Convex relaxation such as $L_1$-regularization that solves the problem under some conditions. However it often leads to sub-optimal sparsity in reality. This paper tries to remedy the above gap between theory and practice. In particular, we investigate a multi-stage convex relaxation scheme for solving problems with non-convex regularization. Theoretically, we analyze the behavior of a resulting two-stage relaxation scheme for the capped-$L_1$ regularization. Our performance bound shows that the procedure is superior to the standard $L_1$ convex relaxation for learning sparse targets. Experiments confirm the effectiveness of this method on some simulation and real data.

One of the central problems in neuroscience is reconstructing synaptic connectivity in neural circuits. Synapses onto a neuron can be probed by sequentially stimulating potentially pre-synaptic neurons while monitoring the membrane voltage of the post-synaptic neuron. Reconstructing a large neural circuit using such a “brute force” approach is rather time-consuming and inefficient because the connectivity in neural circuits is sparse. Instead, we propose to measure a post-synaptic neuron’s voltage while stimulating simultaneously multiple randomly chosen potentially pre-synaptic neurons. To extract the weights of individual synaptic connections we apply a decoding algorithm recently developed for compressive sensing. Compared to the brute force approach, our method promises significant time savings that grow with the size of the circuit. We use computer simulations to find optimal stimulation parameters and explore the feasibility of our reconstruction method under realistic experimental conditions including noise and non-linear synaptic integration. Multiple-neuron stimulation allows reconstructing synaptic connectivity just from the spiking activity of post-synaptic neurons, even when sub-threshold voltage is unavailable. By using calcium indicators, voltage-sensitive dyes, or multi-electrode arrays one could monitor activity of multiple post-synaptic neurons simultaneously, thus mapping their synaptic inputs in parallel, potentially reconstructing a complete neural circuit.

A non-parametric Bayesian model is proposed for processing multiple images. The analysis employs image features and, when present, the words associated with accompanying annotations. The model clusters the images into classes, and each image is segmented into a set of objects, also allowing the opportunity to assign a word to each object (localized labeling). Each object is assumed to be represented as a heterogeneous mix of components, with this realized via mixture models linking image features to object types. The number of image classes, number of object types, and the characteristics of the object-feature mixture models are inferred non-parametrically. To constitute spatially contiguous objects, a new logistic stick-breaking process is developed. Inference is performed efficiently via variational Bayesian analysis, with example results presented on two image databases.

An algorithm is presented for online learning of rotations. The proposed algorithm involves matrix exponentiated gradient updates and is motivated by the von Neumann divergence.The multiplicative updates are exponentiated skew-symmetric matrices which comprise the Lie algebra of the rotation group. The orthonormality andunit determinant of the matrix parameter are preserved using matrix logarithms andexponentials and the algorithm lends itself to intuitive interpretation in terms of the differential geometry of the manifold associated with the rotation group. A complexity reduction result is presented that exploits the eigenstructure of the matrix updates to simplify matrix exponentiation to a quadratic form.

Learning in real-time applications, e.g., online approximation of the inverse dynamics model for model-based robot control, requires fast online regression techniques. Inspired by local learning, we propose a method to speed up standard Gaussian Process regression (GPR) with local GP models (LGP). The training data is partitioned in local regions, for each an individual GP model is trained. The prediction for a query point is performed by weighted estimation using nearby local models. Unlike other GP approximations, such as mixtures of experts, we use a distance based measure for partitioning of the data and weighted prediction. The proposed method achieves online learning and prediction in real-time. Comparisons with other nonparametric regression methods show that LGP has higher accuracy than LWPR and close to the performance of standard GPR and nu-SVR.

We present an approach to low-level vision that combines two main ideas: the use of convolutional networks as an image processing architecture and an unsupervised learning procedure that synthesizes training samples from specific noise models. We demonstrate this approach on the challenging problem of natural image denoising. Using a test set with a hundred natural images, we find that convolutional networks provide comparable and in some cases superior performance to state of the art wavelet and Markov random field (MRF) methods. Moreover, we find that a convolutional network offers similar performance in the blind denoising setting as compared to other techniques in the non-blind setting. We also show how convolutional networks are mathematically related to MRF approaches by presenting a mean field theory for an MRF specially designed for image denoising. Although these approaches are related, convolutional networks avoid computational difficulties in MRF approaches that arise from probabilistic learning and inference. This makes it possible to learn image processing architectures that have a high degree of representational power (we train models with over 15,000 parameters), but whose computational expense is significantly less than that associated with inference in MRF approaches with even hundreds of parameters.

In this paper, we study the manifold regularization for the Sliced Inverse Regression (SIR). The manifold regularization improves the standard SIR in two aspects: 1) it encodes the local geometry for SIR and 2) it enables SIR to deal with transductive and semi-supervised learning problems. We prove that the proposed graph Laplacian based regularization is convergent at rate root-n. The projection directions of the regularized SIR are optimized by using a conjugate gradient method on the Grassmann manifold. Experimental results support our theory.

We present AROW, a new online learning algorithm that combines several properties of successful : large margin training, confidence weighting, and the capacity to handle non-separable data. AROW performs adaptive regularization of the prediction function upon seeing each new instance, allowing it to perform especially well in the presence of label noise. We derive a mistake bound, similar in form to the second order perceptron bound, which does not assume separability. We also relate our algorithm to recent confidence-weighted online learning techniques and empirically show that AROW achieves state-of-the-art performance and notable robustness in the case of non-separable data.

Most models of categorization learn categories defined by characteristic features but some categories are described more naturally in terms of relations. We present a generative model that helps to explain how relational categories are learned and used. Our model learns abstract schemata that specify the relational similarities shared by instances of a category, and our emphasis on abstraction departs from previous theoretical proposals that focus instead on comparison of concrete instances. Ourfirst experiment suggests that abstraction can help to explain some of the findings that have previously been used to support comparison-based approaches. Oursecond experiment focuses on one-shot schema learning, a problem that raises challenges for comparison-based approaches but is handled naturally by our abstraction-based account.